One can find many suitable thresholds from the bandit literature [1]}, [2]}, [3]}, all of which are of the order \(\beta (\delta ,t) \approx \log (1/\delta ) + \frac{K}{2} \log (\log (t/\delta )) \)  is enough to ensure that \(\mathbb {P}\big (G_\varepsilon (\widehat{{\mu }}_{\tau _\delta }) \ne G_\varepsilon ({\mu })\big ) \le \delta \) , i.e. that the algorithm is \(\delta \) -correct.
